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problem #280 - posted monday, november 3, 2008
:. first problem of season 12 :.

 
? Less , Equal , Greater ?
That should cover it, at least for real numbers...

 
i) Suppose a + b + c = 1 , where a , b , and c are non-neg. real nos.
Prove that we always have : 7(a b + a c + b c) <= 2 + 9 a b c .
 
ii) Now suppose that for all reals 0 <= a <= b <= c <= d , we have :
(a + b + c + d)^2 >= K b c . Find the largest possible value of K.
 

Show your steps, formulas, and reasoning; one point penalty for resubmissions.
.
 
deadline is tuesday, nov 11, 2008 by Midnight Pacific Time USA
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(c) 1997-2008 Dan Bach and B & L Math Enterprises; all rights reserved. Download for personal use only.
 
 
 
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Current Year 2007-08 . . . . All-Time Leader Board
 
Final Standings of past years: on a separate page :
 
2006/07 . 2005-06 . 2004-05 . 2003-04 . 2002-03
 
2001-02 . 2000/01 . 1999/2000 . 1998/99 . 1997/98
 

Year 11 - Current 2007/08 STANDINGS - 7 problems (#269-275)
Oct.'07 - Oct.'08 contest (all-time standings below, previous years above )
This week's entrants in blue, recent in purple (last week's ranking in parentheses)
 
1.Nick McGrath (1) . . . . . 40
2. Moreau deSaint-Martin (2) . . . 32
3. Ed Wern (4) . . . . . . . . . . . . . . . 30
4. Tim Poe (4) . . . . . . . . . . . . . . . 28
5. Etienne Desclin (4) . . . . . . . . . 26
6. Philippe Fondanaiche (8) . . . 24
7. Nats Kroy (3). . . . . . . . . . . . . . 23
8. Joe Fendel (4) . . . . . . . . . . . . . 22
9. Claudio Baiocchi (9) . . . . . . . 21
10. Art Morris (11) . . . . . . . . . . . 19
 
11. James Laverty (11) . . . . . . . . 14
11. Marcello Cammarata (10) . . 14
13. Allen Druze (16). . . . . . . . . . 11
14. Phil Sayre (19) . . . . . . . . . . . 9
14. Frank Mullin (19) . . . . . . . . 9
14. Kirk Bresniker (13) . . . . . . . 9
14. Mark Rickert (13) . . . . . . . . 9
18. Zahi Teitelman (21) . . . . . . . 7
18. K Sengupta (17) . . . . . . . . . . 7
18. Hermen Jacobs (17) . . . . . . . 7
 
Yakov Macek . . . . . . . . . . 6
Ravi Raja . . . . . . . . . . . . 5
Denis Borris . . . . . . . . . . 5
Bob Chen . . . . . . . . . . . 4
Radu Ionescu . . . . . . . . 4
David Madfes . . . . . . . . 4
Quasi-C. . . . . . . . . . . . . . 4
Bradley Wild . . . . . . . . . 4
 
Earl Gose (new) . . . . . . . . 3
Sarah P. . . . . . . . . . . . . 3
Ondrej Calda . . . . . . . . 3
Garry Malashkin . . . . . 3
Alan O'Donnell. . . . . . . 3
Jin Won Park . . . . . . . 3
Steve Tonnesen. . . . . . . 3
David Stigant. . . . . . . . 3
Dawn Haught. . . . . . . . 3
Mark Moyer . . . . . . . . . 3
JaeKon Lee . . . . . . . . . . 3
 
Ajit Athle . . . . . . . . . . . 2
Riley Wern . . . . . . . . . . 2
Ken Duisenberg . . . . . . 2
Alex Kim. . . . . . . . . . . . 2
Brady Wern . . . . . . . . . 2
Art Wolfskill . . . . . . . . 2
Morandi Maurizio . . . 1
Kyle Misiaszek . . . . . . 1
 
 
 

 
Cumulative All-Time Standings - after 275 problems
This week's entrants in blue, recent in purple (last week's ranking in parentheses)
 
1. Arthur Morris (1) . . . . . 862
2. Nick McGrath (2) . . . . . 822
3. Tim Poe (3) . . . . . . . . . . 693
4. Phil Sayre (4) . . . . . . . . 546
5. Marcello Cammarata (5) . 504
6. Hermen Jacobs (6) . . . . 491
7. Ed Wern (7) . . . . . . . . . 417
8. Quasi-C (8) . . . . . . . . . . 335
9. Lisa Schechner (9) . . . . 315
10. Sue B (10) . . . . . . . . . . 299
 
11. Mark Rickert (11) . . . 289
12. Denis Borris (12) . . . . 263
13. Andy Murdock (13) . . 251
14. Zahi Teitelman (14) . . 236
15. Jeremy Galvagni(15) . 225
16. Allen Druze (16) . . . . . 209
17. Jack Dostal (17) . . . . . 197
18. Vince LoCascio (18) . . 185
19.Philippe Fondanaiche(19) 178
20. Kirk Bresniker (20) . . 170
20. Ludwig Deruyck (20) . 170
 
22. Al B. (22) . . . . . . . . . . . . . . . . 161
23. Steve Lawrie (23) . . . . . . . . . 158
24. Patrik Petersson (24) . . . . . 141
25. Beth Wilson (25) . . . . . . . . . 138
26. Joe Alvord (26) . . . . . . . . . . 136
27. Radu Ionescu (28) . . . . . . . . 135
27. Sudipta Das (27) . . . . . . . . . 135
29. Anirban Bhattacharyya (29) . . 129
30. Claudio Baiocchi (32) . . . . 128
 
31. Nikita Kuznetsov (30) . . . . 127
32. Ajit Athle (31). . . . . . . . . . . 126
33. Alan O'Donnell (33) . . . . . .123
34. Fanfan (34) . . . . . . . . . . . . . 113
35. Akifumi Iwahashi (35) . . . . 108
35. Tim Nelson (35) . . . . . . . . . 108
37. Yakov Macak (37) . . . . . . 106
38. K Sengupta (38) . . . . . . . . .101
39. Nats Kroy (39) . . . . . . . . . . . 97
40. Jayavel Sounderpandian (40) 94
 
41. Jon Stearn (41) . . . . . . . . . 89
42. Etienne Desclin (44) . . . . . 88
43. Prachai K.(42) . . . . . . . . . . 86
44. Drew (43) . . . . . . . . . . . . . . 85
45. Irina Kolesnik (45) . . . . . . 80
46. Steve Carlon (46) . . . . . . . 78
47. Ravi Raja (48) . . . . . . . . . 76
47. Ken Duisenberg (47) . . . . 76
49. Joao Sousa (49). . . . . . . . . 73
50. Filipe Gonçalves (50) . . . . 70
 
Rebecca Wooten . . . . . . . 62
Garry Malashkin . . . . . . 60
Al Nelson . . . . . . . . . . . . . 60
Zeke Moore . . . . . . . . . . . 59
Slanky . . . . . . . . . . . . . . . 56
Trevor Bird. . . . . . . . . . . 56
Gaurav Agrawal . . . . . . 55
Mario Roederer . . . . . . . 53
S. Khoo. . . . . . . . . . . . . . 51
Huan-yu Chung . . . . . . 51
 
Colleen Frye . . . . . . . . 48
Sarah DeSanctis. . . . . . 47
V Balakrishnan. . . . . . 46
Dave Seaman . . . . . . . 45
Kevin Williams. . . . . . 44
Moreau deSaint-Martin . 41
Joe Fendel . . . . . . . . . 40
Oliver Schebanek. . . . 37
Mohamed Omar. . . . . 37
Greg Hess . . . . . . . . . . 35
Shinsuke Kuzuya. . . . 33
 
David Madfes . . . . . . 31
Omar Gymboski . . . . 31
Les Billig. . . . . . . . . . . 31
Danielle Janney. . . . . 30
Chen Sin Tak . . . . . . 29
Mahbubul Hasan . . . 27
Uwe Buenting . . . . . . 27
Derek Chin . . . . . . . . 27
Albert Alvarez . . . . . 24
Carl Chenard . . . . . . 24
Jeffrey Hunt . . . . . . . 24
 
David Stigant . . . . . . 23
Chris Pentacoff. . . . . 23
Andrew Bajunemi. . . 23
James Laverty . . . . . 22
Coyotek . . . . . . . . . . . 22
AZ Runner . . . . . . . . 22
Michelle Lee. . . . . . . . 22
Guha Kaushik. . . . . . 22
Miao. . . =^o^= . . . . . 22
Doug Babcock. . . . . . 21
Rick Montgomery . . 20
Deborah Fradelis . . . 19
Adam Panagos . . . . . 19
Sam Wilson. . . . . . . . 19
 
Nic Hargreaves . . . . .18
Yu Chen Wang. . . . . 18
Juan Carlos Carrara 17
Po-Nien Chen . . . . . . 17
Peter Williams . . . . . 17
Frank Mullin . . . . . . 16
Marcin Mika . . . . . . 16
Big Dave . . . . . . . . . . 16
Giridihar Prasannan 15
Payman Parang. . . . 15
Montta . . . . . . . . . . . 15
Cheryl Lim Xuanzi . 15
 
Jin Won Park . . . . . 14
Thad . . . . . . . . . . . . . 14
Charles Lo . . . . . . . . 14
Tat Tsui. . . . . . . . . . . 14
Ji Zheng . . . . . . . . . . 13
Tim Boomer . . . . . . . 13
Chris North . . . . . . . 13
Jesse Goldlink . . . . . .13
Maria Pereira. . . . . . 13
Jeff O'Connell . . . . . 13
Jason M. McCargar. 13
 
b basic . . . . . . . . . . . . 12
Wanda Cahill . . . . . . 12
Troy Butler . . . . . . . . 12
Paul Botham . . . . . . 12
Pat Gremban . . . . . . 12
Steve Tonnesen. . . . . 11
Farid Mark Watson. 11
Dangkhoa Pham . . . 11
Matt Situ . . . . . . . . . 11
Kia Davidson . . . . . . 11
 
Ameya Joshi . . . . . . . 10
David Huang . . . . . . 10
Abbas Mehrabian . . 10
Navi Sidhu . . . . . . . . 10
Charles Stanton . . . . 10
Matt Rattigan. . . . . . 10
Kenneth Potstada. . . 10
Jen-Kai Pan . . . . . . . 10
James Layland. . . . . 10
Dave B. . . . . . . . . . . . 10
 
Dennis Molter . . . . . . . . 9
Yonlawan Chirawatcharadej 9
Rajwant Singh. . . . . . . . 9
Joon Sup. . . . . . . . . . . . . 9
 
Morandi Maurizio. . . . . 8
Wenjun Kang. . . . . . . . . 8
Monika Reynolds . . . . . 8
Ritwik Chaudhuri . . . . 8
Tim Bock . . . . . . . . . . . . 8
Katherine Dostal . . . . . . 8
Alex Chan . . . . . . . . . . . 8
Sabina Paradi . . . . . . . . 8
Punam Bhullar . . . . . . . 8
Nilesh Patel . . . . . . . . . . 8
Sarah Mahdavi . . . . . . . 8
 
Aclaira . . . . . . . . . . . . . . 7
Joe Schank. . . . . . . . . . . 7
Mark Moyer . . . . . . . . . . 7
Tom Esser . . . . . . . . . . . . 7
Tom Falbo. . . . . . . . . . . . 7
Kelly K. . . . . . . . . . . . . . . 7
Bernie Leitner. . . . . . . . . 7
Wilson Matheson . . . . . . 7
Jonathan Ediyanto. . . . . 7
 
Bob Chen . . . . . . . . . . . . 6
Mark Moyer . . . . . . . . . . 6
Brady Wern . . . . . . . . . . 6
Adam Morgan . . . . . . . . 6
Sara Elsnick. . . . . . . . . . 6
Michael Thwaites. . . . . . 6
Chong Zhang . . . . . . . . . 6
Justin Rowsell . . . . . . . . 6
Charleen Yen . . . . . . . . . 6
Martin Gritsch. . . . . . . . 6
Greg McWhirter. . . . . . . 6
Robert Hussey . . . . . . . . 6
Bruce. . . . . . . . . . . . . . . . 6
Kristen Hehr . . . . . . . . . 6
Monsan . . . . . . . . . . . . . 6
Miho Koto . . . . . . . . . . . 6
Brian Lin . . . . . . . . . . . . 6
Melina Hidayat . . . . . . . 6
Jessica Bubb. . . . . . . . . . 6
Maurice Turrieta. . . . . . 6
George Caleodis. . . . . . . 6
Flavio Cherarola . . . . . . 6
 
Haley Buxton. . . . . . . . . 5
Danny Roe. . . . . . . . . . . 5
Luis deBrito Camacho . 5
Eric Targowski. . . . . . . 5
Sumita Das. . . . . . . . . . 5
Mike Kosciuk. . . . . . . . 5
Robert Dostal . . . . . . . . 5
Jim No-Spam . . . . . . . . 5
Richard Hillman . . . . . 5
Th. Sarojkumar . . . . . . 5
Erin Keithley . . . . . . . . 5
Cindy Tang . . . . . . . . . 5
Richard Clarke . . . . . . 5
Amy Maris. . . . . . . . . . 5
Marowen Ng . . . . . . . . 5
Megan Spillane . . . . . 5
Ileen Huh . . . . . . . . . . 5
Esther Li. . . . . . . . . . . 5
Andy Harmon . . . . . . 5
Vvalker Kellogg. . . . . 5
Connie C. . . . . . . . . . . 5
Heather Jameson . . . . 5
 
Bradley Wild . . . . . . . 4
Jeff Peterson . . . . . . . . 4
Damon Haught. . . . . . 4
Subba Ready . . . . . . . 4
Cliff Angle . . . . . . . . . 4
Gordon Berg . . . . . . . . 4
T Miller . . . . . . . . . . . . 4
Joel Haywood . . . . . . . 4
Shanthy Krishnamurthy . 4
(S.G.) Roy . . . . . . . . . . . 4
Albert Heinrich . . . . . 4
Sandy Thompson . . . . 4
Leora . . . . . . . . . . . . . . 4
Basem Chaikhouni . . 4
Anne Sufka . . . . . . . . . 4
Andy Hsieh . . . . . . . . . 4
Adam Levy . . . . . . . . . 4
Yvette Bitar . . . . . . . . . 4
Tu Truong. . . . . . . . . . 4
Cheryl Fredricks. . . . . 4
Lorenzo Grespan . . . . 4
Max Mojab . . . . . . . . . 4
Samuel McArthur . . . 4
Yuri Sumnicht . . . . . . 4
Nicole Hetzer . . . . . . . 4
Aaron Mikuni . . . . . . 4
Matt Whitlow . . . . . . 4
Cliff Young. . . . . . . . . 4
Kevin Marshall . . . . . 4
CRM . . . . . . . . . . . . . . 4
 
Earl Gose (new) . . . . . . . 3
Sarah P. . . . . . . . . . . . 3
Ondrej Calda . . . . . . . 3
Dawn Haught . . . . . . . 3
JaeKon Lee . . . . . . . . . 3
Kyle Misiaszek . . . . . 3
Arijit Bhattacharyya . . 3
Tony Wang . . . . . . . . . 3
Mo Money . . . . . . . . . . 3
Stefan Gatachiu . . . . . 3
Ameya Mannikar . . . . 3
Abhijit Parashar . . . . 3
Hemingway. . . . . . . . . 3
Felix Chan. . . . . . . . . . 3
Michael Callahan . . . . 3
John Smith . . . . . . . . . 3
Nemesis Enforcer . . . . 3
Arijit Bhattacharyya. . . 3
Charlene Bong . . . . . . 3
Bob Seegmiller . . . . . . 3
Larry Corrado . . . . . . 3
Jerome Mahadeo . . . . 3
Ayane Buccini . . . . . . 3
OsmTechie . . . . . . . . . 3
Dan Carey . . . . . . . . . 3
Marc Wallace . . . . . . 3
Jeff Limber . . . . . . . . 3
Crystal Lai . . . . . . . . 3
Lindsay Dann . . . . . . 3
Myung Hwa Yang . . . 3
Chris Jones . . . . . . . . 3
Uri Andrews . . . . . . . 3
Sukhmine Bains . . . . 3
Tony Sun . . . . . . . . . . 3
Adel Farahmand . . . . 3
Gleb Podkolzin. . . . . 3
Willette . . . . . . . . . . . 3
Cristina Cancio . . . . 3
Yehuda Levy . . . . . . . 3
Martin Gritch . . . . . . 3
John DeLaurentis . . 3
George Perry . . . . . . . 3
Alison Muñoz . . . . . . 3
Joya Guha . . . . . . . . . 3
Richard Miller . . . . . 3
TripleVCourtois . . . . 3
Greg Silin . . . . . . . . . 3
Joseph Hendricks . . . 3
Carl Main . . . . . . . . . 3
Gohan . . . . . . . . . . . . 3
Daniel Cho . . . . . . . . 3
Warren Peace . . . . . . 3
Arthur Thung . . . . . . 3
Greg Vitan . . . . . . . . . 3
Hoodin Hamidi . . . . . 3
 
Riley Wern . . . . . . . . 2
Alex Kim . . . . . . . . . . 2
Art Wolfskill . . . . . . . 2
Bengt Blomster . . . . . 2
Oxana Urdaneta . . . . 2
Jacket . . . . . . . . . . . . . 2
Isha Kotecha. . . . . . . . 2
Keith Downey. . . . . . . 2
Adam Ramage . . . . . . 2
Justin Bradley . . . . . . 2
David B. . . . . . . . . . . . 2
Sean Allaband . . . . . . 2
Rewon C . . . . . . . . . . . 2
Vallee Suennen . . . . . 2
Florian Fischer. . . . . . 2
Dan Greene . . . . . . . . . 2
Charlie Acquisio . . . . 2
Jason McConnell . . . . 2
Barry Thompson . . . . 2
Joe Skolnik . . . . . . . . . 2
Rich Futrell. . . . . . . . . 2
Peter Hendriks . . . . . . 2
Ben Cesch . . . . . . . . . . 2
Jean-Baptiste Brochard 2
Heesoo Chang. . . . . . . . 2
I.M. Fahim . . . . . . . . . . 2
Pascal Picard . . . . . . . . 2
Dawn Snyder . . . . . . . . 2
--The Bozzball . . . . . . . 2+
Keith Anker . . . . . . . . . 2
James Wan. . . . . . . . . . 2
Santo . . . . . . . . . . . . . . 2
Andy Oats . . . . . . . . . . 2
David Nadler . . . . . . . . 2
Anandi (Rao) . . . . . . . . 2
Nick Sherrard . . . . . . . 2
Mahul Bhattacharya . 2
Pam Deutsch . . . . . . . 2
Gilad Skyte . . . . . . . . 2
Andy Oats. . . . . . . . . . 2
Joanne Dunlap . . . . . 2
Ravi Subramanian . . 2
Nemo Dumple . . . . . . 2
Bugz Podder. . . . . . . . 2
Erika Brandner . . . . . 2
Trevor Wells . . . . . . . 2
Paul Robinson . . . . . . 2
Ahearn Bizmah . . . . . 2
Marcel Birthelmer . . . 2
Musthaffa Kannipoyil 2
Jorge Ramirez. . . . . . . 2
Ernesto von Ruckert. . 2
Tai Dang . . . . . . . . . . 2
Mehdi Ansari . . . . . . 2
Mike Fuqua . . . . . . . . 2
Hareenda Yalamanchili 2
Michael Browell. . . . . 2
Jon Wilson. . . . . . . . . 2
Mercedes Foster . . . . . 2
Edwin Wijono . . . . . . 2
Conrad Tan . . . . . . . . 2
Daria Eiteneer. . . . . . 2
Matt Albrecht . . . . . . 2
Dan Lowenthal . . . . . 2
Default . . . . . . . . . . . . 2
P Ranger . . . . . . . . . . 2
John Trott . . . . . . . . 2
Katie Nechayev. . . . . 2
Jaewon Yoon . . . . . . 2
Exotic Nut . . . . . . . . 2
Branden Bedwell . . . 2
Kevin Ho . . . . . . . . . . 2
Quoctan Nguyen. . . . 2
Paige Campbell :-) . . 2
Caroline Hung . . . . . 2
Jenee Donner . . . . . . 2
Tim Gavin . . . . . . . . 2
A. Anaya . . . . . . . . . 2
Reyy . . . . . . . . . . . . . 2
Ray & Dina Leduc . 2
Truman Lee . . . . . . . 2
Tom DeSanctis . . . . . 2
Alix Erie . . . . . . . . . . 2
Camilo Jimenez. . . . 2
Nikki Carr . . . . . . . . 2
Triono Dawis . . . . . . 2
Chris Block . . . . . . . 2
AD/Marsha Reichle . 2
Kristy Poindexter . . . 2
Lisa Cassidy . . . . . . . 2
Aaron Murdock. . . . . 2
Candace Brant. . . . . . 2
 
Mary Noggin . . . . . . . 1
Hendrik van Eijsden 1
Mahul Bhattacharya. 1
Pee Wee . . . . . . . . . . . . 1
Jeff Olson . . . . . . . . . . 1
Polly Lau . . . . . . . . . . 1
Corey Ramsey . . . . . . 1
LetsGoCubs . . . . . . . . 1
Barb Owca . . . . . . . . . 1
Henry Yang. . . . . . . . . 1
Ralph Nguyen . . . . . . 1
Daniel Baczkowski . . 1
Debbie Miller . . . . . . 1`
Divyanshu Mishra . . 1
Abbey the Mathematician 1
Marisa Carcallas . . . 1
Barb G. . . . . . . . . . . . 1
Math 012378 . . . . . . 1
John Friend . . . . . . . 1
vze4bwtf . . . . . . . . . . 1
Wait . . . . . . . . . . . . . 1
Renee Prasad . . . . . . 1
Kathy Serbin . . . . . . 1
Dodopotpie . . . . . . . . 1
Hyunshin Cho . . . . . 1
Fabio Garcia . . . . . . 1
Padmaja . . . . . . . . . . 1
Alex Shen . . . . . . . . . 1
Cyril Virtusio . . . . . 1
Erudite 1A . . . . . . . . 1
Dan Baczkowski. . . . 1
Manuel Fulea . . . . . . 1
Paul Langford. . . . . . 1
Aimee Jacot . . . . . . . . 1
Anthony Leung. . . . . 1
Cari Richardson. . . . 1
Anne Wood. . . . . . . . 1
Brita Groves . . . . . . . 1
Greg Hoggarth . . . . . 1
Monique S. . . . . . . . . 1
Ji-Hong Min . . . . . . . 1
Mary Jane Santos . . . 1
Michael Williams. . . 1
Judy Kane. . . . . . . . . 1
XCute . . . . . . . . . . . . 1
L Mac. . . . . . . . . . . . . 1
Kimy Leung . . . . . . . 1
Gene Oldfield . . . . . . 1
B.M.W. . . . . . . . . . . . 1
Leo S. Bach. . . . . . . . 1
Kavita Aggarwal . . . 1
Randy Toy . . . . . . . . 1
T. Jackowski. . . . . . . 1
Pooboy . . . . . . . . . . . . 1
Zheng Shuwei. . . . . . 1
Rebecca Frye . . . . . . . 1
Kiyomi Sakata . . . . . 1
Utku Aksu . . . . . . . . 1
Erik Thompson. . . . . 1
Daniel Bonilla. . . . . . 1
Mike Birnbaum. . . . . 1
Cathy Bird . . . . . . . . 1
     
Previous Contest Standings 1997-2007
 
 
 
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Deadline for Problem #278 is Mon Sept 22, 2008 !
 
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